An algebra problem by Christopher Boo

Algebra Level 2

\[ \begin{array}{ c c c c c c c c c } && 0. & 1& 1& 1& 1& 1& 1& 1&1& 1&\ldots \\ \times & & 0. & 1& 1& 1& 1& 1& 1& 1&1& 1&\ldots \\ \hline &&0. &0 & 1 & 2& 3& 4& 5& 6& 7& {\color{red}{x}}&\ldots \end{array} \]

The above is a long multiplication, where only the first few digits of each number are shown. (The first two numbers continue repeating 1s.)

What digit is \(x\)?

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